Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3×C7 · Schur multiplier (H2) = C24 |
| direct product | C3×C7×#540 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3×C7 | C24 |
| commutator, Frattini, cosocle | C22 | C24×C3×C7 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C7 ◁ G | |
480 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 672 |
A minimal faithful permutation representation — 4 generators on 26 points:
(1,2,3)(4,5,6,7,8,9,10)(11,12)(13,17)(14,15)(16,18)(20,23)(21,25);(12,15)(16,17)(19,20,22,23)(21,24,25,26);(11,12)(13,16)(14,15)(17,18)(19,20)(21,24)(22,23)(25,26);(11,13,14,18)(12,16,15,17)(19,21,22,25)(20,24,23,26)462 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 11249);